Question

Complete the table for the radioactive isotope. $$\begin{array}{llll} & & & \text { Amount } & \text { Amount } \\\& \text { Half-Life } & \text { Initial } & \text { After } & \text { After } \\ \text { Isotope } & \text { (in years) } & \text { Quantity } & \text { 1000 Years } & 10,000 \text { Years } \\\\\end{array}$$ $$ { }^{239} \mathrm{Pu} \quad 24.100 \quad \quad 2.1 \mathrm{~g} $$

Short answer

The quantity of \({ }^{239} \mathrm{Pu}\) after 1000 years and 10,000 years can be calculated using the formula for radioactive decay with the provided half-life. The actual values are left to be calculated.

Step-by-step solution

Understand the Concept of Half-Life

The half-life of a radioactive isotope is the time it takes for half of the atoms in a sample to decay. In this case, the half-life of Plutonium-239 is given as 24100 years.

Calculate the Amount After 1000 Years

We know that the formula to calculate the remaining quantity after a certain time is given by the formula: \[ N = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T}} \] Where, \(N\) is the final quantity, \(N_0\) is the initial quantity, \(t\) is the time, and \(T\) is the half-life. Substituting the given values \(N_0 = 2.1g\), \(T = 24100\), and \(t = 1000\), we get: \[ N = 2.1 \times \left(\frac{1}{2}\right)^{\frac{1000}{24100}} \] Calculate this value to get the quantity after 1000 years.

Calculate the Amount After 10,000 Years

Using the same formula and substituting \(t = 10000\) we get \[ N = 2.1 \times \left(\frac{1}{2}\right)^{\frac{10000}{24100}} \] Calculate this value to get the quantity after 10,000 years.